Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics
Abstract: Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle problems in the classical domain. In this work, we focus on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes and of generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bit-wise difference between the rules and whether one is much more likely than the other.
- OpenAI, https://chat.openai.com/chat (2024).
- M. Schuld, I. Sinayskiy, and F. Petruccione, An introduction to quantum machine learning, Contemporary Physics 56, 172 (2015).
- S. Cheng, L. Wang, and P. Zhang, Supervised learning with projected entangled pair states, Phys. Rev. B 103, 125117 (2021).
- X. Shi, Y. Shang, and C. Guo, Clustering using matrix product states, Phys. Rev. A 105, 052424 (2022).
- E. M. Stoudenmire, Learning relevant features of data with multi-scale tensor networks, Quantum Science and Technology 3, 034003 (2018).
- L. Deng and Y. Liu, Deep learning in natural language processing (Springer, 2018).
- A. Novikov, M. Trofimov, and I. Oseledets, Exponential machines (2017), arXiv:1605.03795 [stat.ML] .
- V. Pestun, J. Terilla, and Y. Vlassopoulos, Language as a matrix product state (2017), arXiv:1711.01416 [cs.CL] .
- T. Felser, S. Notarnicola, and S. Montangero, Efficient tensor network ansatz for high-dimensional quantum many-body problems, Phys. Rev. Lett. 126, 170603 (2021).
- U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
- I. P. McCulloch, From density-matrix renormalization group to matrix product states, Journal of Statistical Mechanics: Theory and Experiment 2007, P10014 (2007).
- G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
- U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue.
- M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Communications in mathematical physics 144, 443 (1992).
- R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of physics 349, 117 (2014).
- S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys. 55, 601 (1983).
- C. Guo, K. Modi, and D. Poletti, Tensor-network-based machine learning of non-markovian quantum processes, Phys. Rev. A 102, 062414 (2020).
- P.-Y. Louis and F. Nardi, Probabilistic Cellular Automata : Theory, Applications and Future Perspectives (2018).
- M. Cook, Universality in elementary cellular automata, Complex Systems 15 (2004).
- https://www.nscc.sg .
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